\(\int \frac {\arctan (a x)^3}{x^3 (c+a^2 c x^2)} \, dx\) [394]

Optimal result

Integrand size = 22, antiderivative size = 262 \[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )} \, dx=-\frac {3 i a^2 \arctan (a x)^2}{2 c}-\frac {3 a \arctan (a x)^2}{2 c x}-\frac {a^2 \arctan (a x)^3}{2 c}-\frac {\arctan (a x)^3}{2 c x^2}+\frac {i a^2 \arctan (a x)^4}{4 c}+\frac {3 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {3 i a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c}+\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c}-\frac {3 a^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c}-\frac {3 i a^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{4 c} \]

[Out]

-3/2*I*a^2*arctan(a*x)^2/c-3/2*a*arctan(a*x)^2/c/x-1/2*a^2*arctan(a*x)^3/c-1/2*arctan(a*x)^3/c/x^2+1/4*I*a^2*a
rctan(a*x)^4/c+3*a^2*arctan(a*x)*ln(2-2/(1-I*a*x))/c-a^2*arctan(a*x)^3*ln(2-2/(1-I*a*x))/c-3/2*I*a^2*polylog(2
,-1+2/(1-I*a*x))/c+3/2*I*a^2*arctan(a*x)^2*polylog(2,-1+2/(1-I*a*x))/c-3/2*a^2*arctan(a*x)*polylog(3,-1+2/(1-I
*a*x))/c-3/4*I*a^2*polylog(4,-1+2/(1-I*a*x))/c

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5038, 4946, 5044, 4988, 2497, 5004, 5112, 5116, 6745} \[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 c}-\frac {3 a^2 \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{2 c}+\frac {i a^2 \arctan (a x)^4}{4 c}-\frac {a^2 \arctan (a x)^3}{2 c}-\frac {3 i a^2 \arctan (a x)^2}{2 c}-\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {3 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {3 i a^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 c}-\frac {3 i a^2 \operatorname {PolyLog}\left (4,\frac {2}{1-i a x}-1\right )}{4 c}-\frac {\arctan (a x)^3}{2 c x^2}-\frac {3 a \arctan (a x)^2}{2 c x} \]

[In]

Int[ArcTan[a*x]^3/(x^3*(c + a^2*c*x^2)),x]

[Out]

(((-3*I)/2)*a^2*ArcTan[a*x]^2)/c - (3*a*ArcTan[a*x]^2)/(2*c*x) - (a^2*ArcTan[a*x]^3)/(2*c) - ArcTan[a*x]^3/(2*
c*x^2) + ((I/4)*a^2*ArcTan[a*x]^4)/c + (3*a^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c - (a^2*ArcTan[a*x]^3*Log[2
- 2/(1 - I*a*x)])/c - (((3*I)/2)*a^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c + (((3*I)/2)*a^2*ArcTan[a*x]^2*PolyLog
[2, -1 + 2/(1 - I*a*x)])/c - (3*a^2*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c) - (((3*I)/4)*a^2*PolyLog
[4, -1 + 2/(1 - I*a*x)])/c

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 4946

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^
n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

Rule 4988

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[(a + b*ArcTan[c*x])
^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Dist[b*c*(p/d), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))
]/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rule 5038

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d,
Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Dist[e/(d*f^2), Int[(f*x)^(m + 2)*((a + b*ArcTan[c*x])^p/(d + e*x
^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]

Rule 5044

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*d*(p + 1))), x] + Dist[I/d, Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]

Rule 5112

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[I*(a + b*ArcTa
n[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x] - Dist[b*p*(I/2), Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[2, 1 - u]
/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - 2*(I
/(I + c*x)))^2, 0]

Rule 5116

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*(
a + b*ArcTan[c*x])^p*(PolyLog[k + 1, u]/(2*c*d)), x] + Dist[b*p*(I/2), Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLo
g[k + 1, u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (
1 - 2*(I/(I + c*x)))^2, 0]

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )} \, dx\right )+\frac {\int \frac {\arctan (a x)^3}{x^3} \, dx}{c} \\ & = -\frac {\arctan (a x)^3}{2 c x^2}+\frac {i a^2 \arctan (a x)^4}{4 c}+\frac {(3 a) \int \frac {\arctan (a x)^2}{x^2 \left (1+a^2 x^2\right )} \, dx}{2 c}-\frac {\left (i a^2\right ) \int \frac {\arctan (a x)^3}{x (i+a x)} \, dx}{c} \\ & = -\frac {\arctan (a x)^3}{2 c x^2}+\frac {i a^2 \arctan (a x)^4}{4 c}-\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {(3 a) \int \frac {\arctan (a x)^2}{x^2} \, dx}{2 c}-\frac {\left (3 a^3\right ) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx}{2 c}+\frac {\left (3 a^3\right ) \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c} \\ & = -\frac {3 a \arctan (a x)^2}{2 c x}-\frac {a^2 \arctan (a x)^3}{2 c}-\frac {\arctan (a x)^3}{2 c x^2}+\frac {i a^2 \arctan (a x)^4}{4 c}-\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c}+\frac {\left (3 a^2\right ) \int \frac {\arctan (a x)}{x \left (1+a^2 x^2\right )} \, dx}{c}-\frac {\left (3 i a^3\right ) \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c} \\ & = -\frac {3 i a^2 \arctan (a x)^2}{2 c}-\frac {3 a \arctan (a x)^2}{2 c x}-\frac {a^2 \arctan (a x)^3}{2 c}-\frac {\arctan (a x)^3}{2 c x^2}+\frac {i a^2 \arctan (a x)^4}{4 c}-\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c}-\frac {3 a^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c}+\frac {\left (3 i a^2\right ) \int \frac {\arctan (a x)}{x (i+a x)} \, dx}{c}+\frac {\left (3 a^3\right ) \int \frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{2 c} \\ & = -\frac {3 i a^2 \arctan (a x)^2}{2 c}-\frac {3 a \arctan (a x)^2}{2 c x}-\frac {a^2 \arctan (a x)^3}{2 c}-\frac {\arctan (a x)^3}{2 c x^2}+\frac {i a^2 \arctan (a x)^4}{4 c}+\frac {3 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c}-\frac {3 a^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c}-\frac {3 i a^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{4 c}-\frac {\left (3 a^3\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c} \\ & = -\frac {3 i a^2 \arctan (a x)^2}{2 c}-\frac {3 a \arctan (a x)^2}{2 c x}-\frac {a^2 \arctan (a x)^3}{2 c}-\frac {\arctan (a x)^3}{2 c x^2}+\frac {i a^2 \arctan (a x)^4}{4 c}+\frac {3 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {3 i a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c}+\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c}-\frac {3 a^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c}-\frac {3 i a^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{4 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.72 \[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {i a^2 \left (\pi ^4-96 \arctan (a x)^2+\frac {96 i \arctan (a x)^2}{a x}+\frac {32 i \left (1+a^2 x^2\right ) \arctan (a x)^3}{a^2 x^2}-16 \arctan (a x)^4+64 i \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )-192 i \arctan (a x) \log \left (1-e^{2 i \arctan (a x)}\right )-96 \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-96 \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )+96 i \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )\right )}{64 c} \]

[In]

Integrate[ArcTan[a*x]^3/(x^3*(c + a^2*c*x^2)),x]

[Out]

((I/64)*a^2*(Pi^4 - 96*ArcTan[a*x]^2 + ((96*I)*ArcTan[a*x]^2)/(a*x) + ((32*I)*(1 + a^2*x^2)*ArcTan[a*x]^3)/(a^
2*x^2) - 16*ArcTan[a*x]^4 + (64*I)*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] - (192*I)*ArcTan[a*x]*Log[1 -
E^((2*I)*ArcTan[a*x])] - 96*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - 96*PolyLog[2, E^((2*I)*ArcTan[
a*x])] + (96*I)*ArcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 48*PolyLog[4, E^((-2*I)*ArcTan[a*x])]))/c

Maple [A] (verified)

Time = 91.86 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.68

method	result	size
derivativedivides	\(a^{2} \left (-\frac {6 i \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {\arctan \left (a x \right )^{2} \left (-i \arctan \left (a x \right )-3 i a x +x \arctan \left (a x \right ) a \right ) \left (a x +i\right )}{2 c \,a^{2} x^{2}}+\frac {3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {\arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {3 i \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {\arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )}{c}-\frac {6 i \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {3 i \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {i \arctan \left (a x \right )^{4}}{4 c}+\frac {3 \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {3 i \arctan \left (a x \right )^{2}}{c}+\frac {3 \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )}{c}\right )\)	\(441\)
default	\(a^{2} \left (-\frac {6 i \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {\arctan \left (a x \right )^{2} \left (-i \arctan \left (a x \right )-3 i a x +x \arctan \left (a x \right ) a \right ) \left (a x +i\right )}{2 c \,a^{2} x^{2}}+\frac {3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {\arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {3 i \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {\arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )}{c}-\frac {6 i \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {3 i \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {i \arctan \left (a x \right )^{4}}{4 c}+\frac {3 \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {3 i \arctan \left (a x \right )^{2}}{c}+\frac {3 \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )}{c}\right )\)	\(441\)

[In]

int(arctan(a*x)^3/x^3/(a^2*c*x^2+c),x,method=_RETURNVERBOSE)

[Out]

a^2*(-6*I/c*polylog(4,(1+I*a*x)/(a^2*x^2+1)^(1/2))-1/2/c*arctan(a*x)^2*(-I*arctan(a*x)-3*I*a*x+x*arctan(a*x)*a
)*(I+a*x)/a^2/x^2+3*I/c*arctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-1/c*arctan(a*x)^3*ln(1-(1+I*a*x)
/(a^2*x^2+1)^(1/2))-3*I/c*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-6/c*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2
+1)^(1/2))+3*I/c*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-1/c*arctan(a*x)^3*ln((1+I*a*x)/(a^2*x^2+
1)^(1/2)+1)-6*I/c*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-6/c*arctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/
2))-3*I/c*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+1/4*I/c*arctan(a*x)^4+3/c*arctan(a*x)*ln(1-(1+I*a*x)/(a^2*x^2
+1)^(1/2))-3*I/c*arctan(a*x)^2+3/c*arctan(a*x)*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1))

Fricas [F]

\[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )} x^{3}} \,d x } \]

[In]

integrate(arctan(a*x)^3/x^3/(a^2*c*x^2+c),x, algorithm="fricas")

[Out]

integral(arctan(a*x)^3/(a^2*c*x^5 + c*x^3), x)

Sympy [F]

\[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{2} x^{5} + x^{3}}\, dx}{c} \]

[In]

integrate(atan(a*x)**3/x**3/(a**2*c*x**2+c),x)

[Out]

Integral(atan(a*x)**3/(a**2*x**5 + x**3), x)/c

Maxima [F]

\[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )} x^{3}} \,d x } \]

[In]

integrate(arctan(a*x)^3/x^3/(a^2*c*x^2+c),x, algorithm="maxima")

[Out]

integrate(arctan(a*x)^3/((a^2*c*x^2 + c)*x^3), x)

Giac [F]

\[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )} x^{3}} \,d x } \]

[In]

integrate(arctan(a*x)^3/x^3/(a^2*c*x^2+c),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^3\,\left (c\,a^2\,x^2+c\right )} \,d x \]

[In]

int(atan(a*x)^3/(x^3*(c + a^2*c*x^2)),x)

[Out]

int(atan(a*x)^3/(x^3*(c + a^2*c*x^2)), x)